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I developed an application that models the trajectory of a light geodesic in the FLWR metric leaving from a galaxy and coming to our. I made for the moment the euclidean case (k=0) with a zero cosmological constant . So, the metric can be written :

[tex]ds^{2}=c^{2}dt^{2}-R(t)^{2}d\sigma^{2} [/tex]

with[tex]

d\sigma^{2}=dr^2+r^2(d\theta^2+sin^2\theta\,d\phi^2)

[/tex]

and[tex]R(t)=R_{0}\,(\frac{3\,H_{0}}{2}\,t)^{2/3} [/tex]

I get differential system from geodesics general equation :

[tex]\frac{d^2\,u^{i}}{ds^2}+\Gamma_{j}_{}^{i}_{k}\,\frac{d\,u^j}{ds}\,\frac{d\,u^k}{ds}=0[/tex]

here is the differential system :

[tex]

\frac{d^{2}ct}{ds^{2}} = -\frac{2}{3}\,R_{0}^{2}\,\bigg(\frac{3\,H_{0}}{2\,c}\bigg)^{4/3}\,(ct)^{1/3}\,\bigg(\bigg(\frac{d\,r}{ds}\bigg)^{2}+r^2\,\bigg(\frac{d\,\theta}{ds}\bigg)^{2}+r^{2}\,sin^{2}(\theta)\,\bigg(\frac{d\,\phi}{ds}\bigg)^{2}\bigg)[/tex]

[tex]\frac{d^{2}r}{ds^{2}} & = & -\frac{4}{3\,ct}\,\frac{d\,ct}{ds}\,\frac{d\,r}{ds}+r\,\bigg(\bigg(\frac{d\,\theta}{ds}\bigg)^{2}-sin^{2}(\theta)\,\bigg(\frac{d\,\phi}{ds}\bigg)^{2}\bigg)[/tex]

[tex]\frac{d^{2}\theta}{ds^{2}} & = & -\frac{2}{r}\,\frac{d\,r}{ds}\,\frac{d\,\theta}{ds}-\frac{4}{3\,ct}\,\frac{d\,ct}{ds}\,\frac{d\,\theta}{ds}+\frac{sin(2\,\theta)}{2}\,\bigg(\frac{d\,\phi}{ds}\bigg)^{2} [/tex]

[tex]\frac{d^{2}\phi}{ds^{2}} & = & -2\,\frac{d\,\phi}{ds}\,\bigg(\frac{2}{3\,ct}\,\frac{d\,ct}{ds}+cotan(\theta)\,\frac{d\,\theta}{ds}+\frac{1}{r}\,\frac{d\,r}{ds}\bigg)[/tex]

I solve this system by Runge-Kutta numerical method. the initial conditions are linked by the following equation :

[tex]\bigg(\frac{d\,ct}{ds}\bigg)_{0}^{2}=R(t_{0})^{2}\bigg(\bigg(\frac{dr}{ds}\bigg)_{0}^{2}+r_{0}^2\bigg(\bigg(\frac{d\theta}{ds}\bigg)_{0}^{2}+sin^{2}(\theta_{0})\bigg(\frac{d\phi}{ds}\bigg)_{0}^{2}\bigg)\bigg)[/tex]

If i take [tex] \frac{d\,\phi}{ds} = \frac{d\,\theta}{ds} = 0 [/tex], ie if i consider only radial trajectories, i get the figure 1 in attachment where i plot the Distance "r" as a function of Local time "ct" expressed in Megaparsec. i took to get this figure :

[tex]\bigg(\frac{d\,r}{ds}\bigg)_{0} = -1 [/tex]

[tex]\bigg(\frac{d\,\theta}{ds}\bigg)_{0} = 0 [/tex]

[tex] \bigg(\frac{d\,\phi}{ds}\bigg)_{0} = 0 [/tex]

[tex] r{0} = 3000 Mpc [/tex]

and

[tex]R_{0} = 1 [/tex]

The dotted line represents the trajectory of light ray and the solid line represents the trajectory of the galaxy where light is emitted.

the problem I encounter is when I'm looking for the propagation of two rays and to compare the actual angular size of the object and the angular size that we observe. This is represented on figure 2 in attachment. I don't know how to make converge the two rays when they arrive in our galaxy. I think that "r" and "theta" should vary simultaneously but what initial conditions should I take to it ?

We can see on figure 3 in attachment the case where curvature k=0. It's possible to have "theta" constant and "r" which decreases but in the two others cases, "theta" and "r" vary in same time. I hope you understand my problem.

Thanks.

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# Light geodesics in FLWR

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